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The objective of this paper is to explore certain properties involving the generalized hypergeometric function ..., as introduced by Khan et al. [27]. Several fractional derivatives and integral formulas involving the generalized hypergeometric function are computed. Further, we derive the solution of generalized fractional kinetic equations involving generalized hypergeometric function. The respective solution is given in terms of Mittag-Leffler function. [ABSTRACT FROM AUTHOR]

The primary object of this article is to introduce (p, q)-beta logarithmic function with extended beta function by using the logarithmic mean. We evaluate different properties and representations of beta logarithmic function. Further, it is evaluated logarithmic distribution, hypergeometric and confluent hypergeometric functions via logarithmic mean are evaluated and their essential properties are studied. Numerous formulas of (p, q)-beta logarithmic functions such as integral formula, derivative formula, transformation formula and generating function are analyzed. [ABSTRACT FROM AUTHOR]

The introduction of two-parameter (p , q) -calculus and Lie algebras in 1991 has spurred a wave of recent research into (p , q) -special polynomials, including (p , q) -Bernoulli, (p , q) -Euler, (p , q) -Genocchi and (p , q) -Frobenius–Euler polynomials. These investigations have been carried out by numerous researchers in order to uncover a wide range of identities associated with these polynomials and applications. In this article, we aim to introduce (p , q) -sine and (p , q) -cosine based λ-array type polynomials and derive numerous properties of these polynomials such as (p , q) -integral representations, (p , q) -partial derivative formulae and (p , q) -addition formulae. It is worth noting that the utilization of the (p , q) -polynomials introduced in this study, along with other (p , q) -polynomials, can lead to the derivation of various identities that differ from the ones presented here. [ABSTRACT FROM AUTHOR]

In this article, we get certain integral representations of the multi-index Wright generalized Bessel function by making use of the extended beta function. This function is presented as a part of the generalized Bessel–Maitland function obtained by taking the extended fractional derivative of the generalized Bessel–Maitland function developed by Özarsalan and Özergin [M. Ali Özarslan and E. Özergin, Some generating relations for extended hypergeometric functions via generalized fractional derivative operator, Math. Comput. Model. 52 2010, 9–10, 1825–1833]. In addition, we demonstrate the exciting connections of the multi-index Wright generalized Bessel function with Laguerre polynomials and Whittaker function. Further, we use the generalized Wright hypergeometric function to calculate the Mellin transform and the inverse of the Mellin transform. [ABSTRACT FROM AUTHOR]

Many authors investigated the characteristics of the Bell, Euler, Bernoulli, and Genocchi polynomials because of their numerous uses in statistics, number theory, and other branches of science. A generating function for mixed-type Apostol–Euler polynomials of order η related with Bell polynomials is presented in this study. We also construct essential identities of Apostol–Euler polynomials of order η connected with Bell polynomials, such as the correlation formula, the implicit summation formula, the derivative formula, some association with Stirling numbers, and specific cases. Also we establish certain symmetric identities and their known consequences. In addition, we investigate several interesting relationships related to umbral calculus. [ABSTRACT FROM AUTHOR]

In this paper, we provide a generating function for mix type Apostol-Genocchi polynomials of order η associated with Bell polynomials. We also derive certain important identities of Apostol Genocchi polynomials of order η based on Bell polynomials, such as the correlation formula, the implicit summation formula, the derivative formula, some correlation with Stirling numbers and their special instances. Moreover, we discover some symmetric identities and their related known results. [ABSTRACT FROM AUTHOR]

A variety of polynomials, their extensions and variants have been extensively investigated, due mainly to their potential applications in diverse research areas. In this paper, we aim to introduce Laguerre-based generalized Apostol type polynomials and investigate some properties and identities involving them. Among them, some differential-recursive relations for the Hermite-Laguerre polynomials, which are expressed in terms of generalized Apostol type numbers and the Laguerre-based generalized Apostol type polynomials, an implicit summation formula and addition-symmetry identities for the Laguerre-based generalized Apostol type polynomials are presented. The results presented here, being very general, are pointed out to reduce to yield some known or new formulas and identities for relatively simple polynomials and numbers. [ABSTRACT FROM AUTHOR]

The main object of this paper is to introduce a new extension of beta function involving generalized Mittag-leffler function and study its important properties, like integral representation, summation formula, derivative formula, beta distribution, transform formula. Using this definition, we introduce new extended hypergeometric and con uent hypergeometric function. [ABSTRACT FROM AUTHOR]

In this paper, we derive some interesting properties for the extended Laguerre-Based Hermite-Bernoulli polynomials. We also establish two symmetric identities and a generating function involving bilateral series for these extended polynomials. Moreover, some special cases of our main results are also pointed out. [ABSTRACT FROM AUTHOR]